On the trace operator for functions of bounded𝔸-variation
Dominic Breit, Lars Diening, Franz Gmeineder
Abstract
We consider the space [math] of functions of bounded [math] -variation. For a given first-order linear homogeneous differential operator with constant coefficients [math] , this is the space of [math] -functions [math] such that the distributional differential expression [math] is a finite (vectorial) Radon measure. We show that for Lipschitz domains [math] , [math] -functions have an [math] -trace if and only if [math] is [math] -elliptic (or, equivalently, if the kernel of [math] is finite-dimensional). The existence of an [math] -trace was previously only known for the special cases that [math] coincides either with the full or the symmetric gradient of the function [math] (and hence covered the special cases BV or BD). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the BV- and BD-settings) but rather compare projections onto the nullspace of [math] as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on [math] .